Suppose X and Y are
Riemann surfaces which have the open unit ball as universal covering surface.
Let dσX,dσY be the hyperbolic metric on X, Y , respectively. Given any
analytic function f : X → Y the principle of hyperbolic metric asserts that
(f∗(dσY)∕dσX)(p) ≦ 1 for each point p ∈ X where f∗(dσY) is the pull-back to X via
f of the hyperbolic metric on Y . Moreover, equality holds if and only if f
is an (unbranched, unlimited) covering of X onto Y . This paper has two
main objectives. The first is to study how the principle of hyperbolic metric
can be strengthened if we only consider analytic functions which are not
coverings. The second is to investigate the set of all analytic coverings of X onto
Y .
